Hey I'm the circumcentre of a right triangle! Do you know where I live?

Geometry Level pending

The circle \(x^2+y^2-4x-4y+4=0\) is inscribed in a triangle which has two of its sides along the coordinate axes.The locus of the circumcentre of the triangle is

\[x+y-xy+k\sqrt{x^2+y^2}\]

The value of \(k\) is?

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